$$
\begin{array}{l}
\int \tan x d x=-\int \frac{1}{\cos x} d(\cos x)=-\ln |\cos x|+c ; \int \cot d x=\int \frac{d(\sin x)}{\sin x}=\ln |\sin x|+c \\
\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\int \frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^{2}}} d\left(\frac{x}{a}\right)=\arcsin \frac{x}{a}+c ; \int \frac{1}{a^{2}+x^{2}} d x=\frac{1}{a} \int \frac{1}{1+\left(\frac{x}{a}\right)^{2}} d\left(\frac{x}{a}\right)=\frac{1}{a} \arctan \frac{x}{a}+c \\
\operatorname{in} 1: \quad \int f[\varphi(x)] \varphi(x) d x=\int f(\varphi(x)) d \varphi(x) \stackrel{\varphi(s)=c}{=} \int f(t) d t=F(t)+c=F[\varphi(x)]+c
\end{array}
$$

![Image](https://open.ocrmath.com/uploads/inset/f69f/f69f04620c219e69eac43f7c490c9ebc_1275.jpg)

Coses: 无理 $\Rightarrow$ 有理(不一责)

例 1: 
$$
\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x=2 \int \sin \sqrt{x} d(\sqrt{x}) \stackrel{\sqrt{x}=t}{=} 2 \int \sin t d t=-2 \cos t+c=-2 \cos \sqrt{x}+c
$$

例2: 
$$
\int \sqrt{\frac{1-x}{1+x}} d x=\int \frac{1-x}{\sqrt{1-x^{2}}} d x=\int \frac{d x}{\sqrt{1-x^{2}}} \leq \int \frac{x}{\sqrt{1-x^{2}}} d x=\arcsin x+\frac{1}{2} \int \frac{d\left(1-x^{2}\right)}{\sqrt{1-x^{2}}}=\arcsin x+\sqrt{1-x^{2}}+c
$$

例3: 
$$
\int \frac{1}{1+\sqrt{x}} d x \stackrel{x=t^{2}}{=} \int \frac{2 t}{1+t} d t=2 \int\left(1-\frac{1}{1+t}\right) d t=2 t-2 \ln || 1+t\right)|+c=2 \sqrt{x}-2 \ln | 1+\sqrt{x} \mid+c
$$

Case 2: 平方和. 差

(1) 
$$
\sqrt{a^{2}-x^{2}} \stackrel{x=a \sin t}{=} a \cos t
$$

倒 
$$
\int \frac{d x}{x^{2}} x=\sin t \quad \cos f d t \quad \frac{\sqrt{a^{2}-x^{2}}}{t} t^{t}
$$

![Image](https://open.ocrmath.com/uploads/inset/f69f/f69f04620c219e69eac43f7c490c9ebc_2362.jpg)

(2) 
$$
\sqrt{x^{2}+a^{2}} \quad x=a \tan t \quad \sin ^{2} t \sin t=\int \csc ^{2} x d t=-\cot t+c=-\frac{\sqrt{1-x^{2}}}{x}+c
$$

(2) 
$$
\sqrt{x^{2}+a^{2}} \stackrel{x=a \tan t}{=} a\sec t \left(1+\tan ^{2} t=\sec ^{2} t\right) = a \int_{x} \sqrt{x^{2}+a^{2}}
$$

例1 
$$
\int \frac{1}{\sqrt{x^{2}+a^{2}}} d x \stackrel{x=a \tan t}{=} \int \frac{a \sec ^{2} t d t}{a \sec t}=\int \sec